All Questions
13 questions
8
votes
1
answer
652
views
Lebesgue entropy zero and positive topological entropy
I am looking for examples of volume preserving $C^{\infty}$ diffeomorphisms $f$ of a surface, which have positive topological entropy ($h(f) > 0$), but that the Lebesgue measure entropy (metric ...
- 1,101
7
votes
1
answer
209
views
Is there a similar theorem in the partially hyperbolic case?
Theorem 5.10.3 from Introduction to dynamical systems, by Brin & Stuck:
Let $f:M\rightarrow M$ be an Anosov diffeomorphism. Then the following are equivalent:
$NW(f)=M$,
every unstable manifold ...
- 279
7
votes
1
answer
281
views
Quantitative approximation of invariant measures by periodic ones
It is well known that many strongly chaotic dynamical systems have the property that periodic measures are (weak-star) dense in the space of all invariant probability measures. Is there some knowledge ...
- 173
6
votes
4
answers
763
views
A follow up question related to entropy
For a self-map $\varphi:X\longrightarrow X$ of a space $X$, many important notions of entropy are defined through a limit of the form $$\lim_{n\rightarrow\infty}\frac{1}{n}\log a_n,$$ where in each ...
- 3,101
4
votes
1
answer
212
views
When entropy SRB measure is zero
It is well known that many strongly chaotic dynamical systems have the property that periodic measures are (weak-star) dense in the space of all invariant probability measures.
Let $f:M \rightarrow ...
- 1,043
2
votes
1
answer
283
views
the union of local stable manifolds along local unstable manifolds
Let $f:M\rightarrow M$ be a $C^2$ hyperbolic diffeomorphism on compact connected riemannian manifold $M$. then there are local stable and unstable manifolds at each point denoted by $W^s_\delta(x), W^...
- 279
2
votes
1
answer
254
views
Chaotic dynamics of maps on unit square that are NOT Triangular
We will denote the compact interval $[0,1]$ by $I$ and the unit square $[0,1]\times[0,1]$ by $I^2$. Triangular map on $I^2$ is a continuous map $F:I^2\to I^2$ of the form $F(x,y)=(f(x),g(x,y))$ where $...
- 271
2
votes
0
answers
319
views
A (possible) generic spectral property in one dimensional dynamics
Context and Definitions
Consider the interval $I=[0,1]$. We say that $T:I\to I$ satisfies the axiom A (I am following [1]) if:
$T$ has a finite number of hyperbolic periodic attractors; and
defining $...
- 333
2
votes
0
answers
124
views
On invariant cones of the Katok map
I am studying the Katok map and similarly constructed examples of nonuniformly hyperbolic surface diffeomorphisms. An important part of the analysis of these diffeomorphisms is the invariance of a ...
- 151
2
votes
0
answers
94
views
Algebraic approach to prove the mixing property of Lorenz flow on hyperbolic surface
We knew that the noncompact subgroups of SL(2,$\mathbb{R}$) are mixing by Howe-Moore ergodicity theorem. I am curious about Lorenz flow, if we have a algebraic approach to prove the mixing property of ...
- 53
1
vote
1
answer
227
views
whether there are some books and original papers ergodic theory approach to ODE
Recently I become more and more interested in the field of ergodic theory, especially in the dimension theory and thermal formalism and its applications.
People always said that most of the ideas in ...
- 1,706
1
vote
1
answer
128
views
Continuity of Lyapunov spaces
The following appears in the paper "Continuity properties of entropy" by Newhouse from 1989:
Let $M$ be some smooth Riemannian compact manifold (you may assume boundary-less), and let $f\in Diff^{1+\...
- 502
1
vote
0
answers
175
views
Example of topologically transitive dynamical system with invariant non-ergodic Borel measure
Let $U \subset M$ be an open subset of a Riemannian manifold. I’m trying to find or construct an example of a topologically transitive dynamical system $f : U \to U$ for which
$f : \Lambda \to \...
- 151